Squares in Partial

We investigate the number of positions that do not start a square, the number of square occurrences, and the number of distinct squares in partial words, i.e., sequences that may have undefined positions called holes. We show that the limit of the ratio of the maximum number of positions not starting a square in a binary partial word with h holes over its length n is 15/31 and the limit of the ratio of the minimum number of square occurrences in a binary partial word with h holes over its length n is 103/187, provided the limit of h/n is 0. Both limits turn out to match with the known limits for binary full words (those without holes). We prove another surprising result that the maximal proportion of defined positions that are square-free to the number of defined positions in a binary partial word with h holes of length n is 1/2, provided the limit of h/n is in the interval [1/11, 1). We also give a 2k tight bound on the number of rightmost occurrences of squares per position in a k-ary partial word with h holes. In addition, we provide a more detailed analysis than earlier ones for the maximum number of distinct squares in a one-hole partial word of length n over an alphabet of size k, bound that is independent of k. ∗This material is based upon work supported by the National Science Foundation under Grant No. DMS–1060775. Part of this paper was presented at DLT 2012 [3]. Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402–6170, USA, [email protected] Department of Mathematical Sciences, Carnegie Mellon University, Wean Hall 6113, Pittsburgh, PA 15213, USA Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, USA Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544–1000, USA